Cubic critical

Percentage Accurate: 52.1% → 86.0%

Time: 14.3s

Alternatives: 9

Speedup: 11.6×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 52.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 86.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.9 \cdot 10^{+153}:\\ \;\;\;\;\left(b \cdot 2\right) \cdot \frac{1}{a \cdot -3}\\ \mathbf{elif}\;b \leq 1.75 \cdot 10^{-63}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.9e+153)
   (* (* b 2.0) (/ 1.0 (* a -3.0)))
   (if (<= b 1.75e-63)
     (/ (- (sqrt (- (* b b) (* (* a 3.0) c))) b) (* a 3.0))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.9e+153) {
		tmp = (b * 2.0) * (1.0 / (a * -3.0));
	} else if (b <= 1.75e-63) {
		tmp = (sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5.9d+153)) then
        tmp = (b * 2.0d0) * (1.0d0 / (a * (-3.0d0)))
    else if (b <= 1.75d-63) then
        tmp = (sqrt(((b * b) - ((a * 3.0d0) * c))) - b) / (a * 3.0d0)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.9e+153) {
		tmp = (b * 2.0) * (1.0 / (a * -3.0));
	} else if (b <= 1.75e-63) {
		tmp = (Math.sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5.9e+153:
		tmp = (b * 2.0) * (1.0 / (a * -3.0))
	elif b <= 1.75e-63:
		tmp = (math.sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5.9e+153)
		tmp = Float64(Float64(b * 2.0) * Float64(1.0 / Float64(a * -3.0)));
	elseif (b <= 1.75e-63)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 3.0) * c))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5.9e+153)
		tmp = (b * 2.0) * (1.0 / (a * -3.0));
	elseif (b <= 1.75e-63)
		tmp = (sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5.9e+153], N[(N[(b * 2.0), $MachinePrecision] * N[(1.0 / N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.75e-63], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 3.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.9000000000000002e153

   1. Initial program 43.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. frac-2neg 43.8%

         \[\leadsto \color{blue}{\frac{-\left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{-3 \cdot a}} \]

      2. div-inv 43.8%

         \[\leadsto \color{blue}{\left(-\left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)\right) \cdot \frac{1}{-3 \cdot a}} \]

   4. Applied egg-rr 43.9%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(c, a \cdot -3, {b}^{2}\right)}\right) \cdot \frac{1}{a \cdot -3}} \]

   5. Taylor expanded in b around -inf 97.7%

      \[\leadsto \color{blue}{\left(2 \cdot b\right)} \cdot \frac{1}{a \cdot -3} \]
   6. Step-by-step derivation

      1. *-commutative 97.7%

         \[\leadsto \color{blue}{\left(b \cdot 2\right)} \cdot \frac{1}{a \cdot -3} \]

   7. Simplified 97.7%

      \[\leadsto \color{blue}{\left(b \cdot 2\right)} \cdot \frac{1}{a \cdot -3} \]

   if -5.9000000000000002e153 < b < 1.75000000000000002e-63

   1. Initial program 79.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   if 1.75000000000000002e-63 < b

   1. Initial program 9.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf 87.3%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. *-commutative 87.3%

         \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]

   5. Simplified 87.3%

      \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.9 \cdot 10^{+153}:\\ \;\;\;\;\left(b \cdot 2\right) \cdot \frac{1}{a \cdot -3}\\ \mathbf{elif}\;b \leq 1.75 \cdot 10^{-63}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.85 \cdot 10^{-35}:\\ \;\;\;\;\left(b \cdot 2\right) \cdot \frac{1}{a \cdot -3}\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{-145}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot -3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.85e-35)
   (* (* b 2.0) (/ 1.0 (* a -3.0)))
   (if (<= b 4.4e-145)
     (/ (- (sqrt (* (* a -3.0) c)) b) (* a 3.0))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.85e-35) {
		tmp = (b * 2.0) * (1.0 / (a * -3.0));
	} else if (b <= 4.4e-145) {
		tmp = (sqrt(((a * -3.0) * c)) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.85d-35)) then
        tmp = (b * 2.0d0) * (1.0d0 / (a * (-3.0d0)))
    else if (b <= 4.4d-145) then
        tmp = (sqrt(((a * (-3.0d0)) * c)) - b) / (a * 3.0d0)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.85e-35) {
		tmp = (b * 2.0) * (1.0 / (a * -3.0));
	} else if (b <= 4.4e-145) {
		tmp = (Math.sqrt(((a * -3.0) * c)) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.85e-35:
		tmp = (b * 2.0) * (1.0 / (a * -3.0))
	elif b <= 4.4e-145:
		tmp = (math.sqrt(((a * -3.0) * c)) - b) / (a * 3.0)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.85e-35)
		tmp = Float64(Float64(b * 2.0) * Float64(1.0 / Float64(a * -3.0)));
	elseif (b <= 4.4e-145)
		tmp = Float64(Float64(sqrt(Float64(Float64(a * -3.0) * c)) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.85e-35)
		tmp = (b * 2.0) * (1.0 / (a * -3.0));
	elseif (b <= 4.4e-145)
		tmp = (sqrt(((a * -3.0) * c)) - b) / (a * 3.0);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.85e-35], N[(N[(b * 2.0), $MachinePrecision] * N[(1.0 / N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.4e-145], N[(N[(N[Sqrt[N[(N[(a * -3.0), $MachinePrecision] * c), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.8499999999999999e-35

   1. Initial program 65.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. frac-2neg 65.4%

         \[\leadsto \color{blue}{\frac{-\left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{-3 \cdot a}} \]

      2. div-inv 65.4%

         \[\leadsto \color{blue}{\left(-\left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)\right) \cdot \frac{1}{-3 \cdot a}} \]

   4. Applied egg-rr 65.5%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(c, a \cdot -3, {b}^{2}\right)}\right) \cdot \frac{1}{a \cdot -3}} \]

   5. Taylor expanded in b around -inf 87.1%

      \[\leadsto \color{blue}{\left(2 \cdot b\right)} \cdot \frac{1}{a \cdot -3} \]
   6. Step-by-step derivation

      1. *-commutative 87.1%

         \[\leadsto \color{blue}{\left(b \cdot 2\right)} \cdot \frac{1}{a \cdot -3} \]

   7. Simplified 87.1%

      \[\leadsto \color{blue}{\left(b \cdot 2\right)} \cdot \frac{1}{a \cdot -3} \]

   if -1.8499999999999999e-35 < b < 4.39999999999999998e-145

   1. Initial program 80.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0 69.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{-3 \cdot \left(a \cdot c\right)}}}{3 \cdot a} \]
   4. Step-by-step derivation

      1. *-commutative 69.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -3}}}{3 \cdot a} \]

      2. *-commutative 69.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(c \cdot a\right)} \cdot -3}}{3 \cdot a} \]

      3. associate-*r* 69.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{c \cdot \left(a \cdot -3\right)}}}{3 \cdot a} \]

   5. Simplified 69.3%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{c \cdot \left(a \cdot -3\right)}}}{3 \cdot a} \]

   if 4.39999999999999998e-145 < b

   1. Initial program 14.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf 82.0%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. *-commutative 82.0%

         \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]

   5. Simplified 82.0%

      \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.85 \cdot 10^{-35}:\\ \;\;\;\;\left(b \cdot 2\right) \cdot \frac{1}{a \cdot -3}\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{-145}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot -3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.6% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.7 \cdot 10^{-247}:\\ \;\;\;\;\left(b \cdot 2\right) \cdot \frac{1}{a \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.7e-247) (* (* b 2.0) (/ 1.0 (* a -3.0))) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.7e-247) {
		tmp = (b * 2.0) * (1.0 / (a * -3.0));
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 2.7d-247) then
        tmp = (b * 2.0d0) * (1.0d0 / (a * (-3.0d0)))
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.7e-247) {
		tmp = (b * 2.0) * (1.0 / (a * -3.0));
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.7e-247:
		tmp = (b * 2.0) * (1.0 / (a * -3.0))
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.7e-247)
		tmp = Float64(Float64(b * 2.0) * Float64(1.0 / Float64(a * -3.0)));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.7e-247)
		tmp = (b * 2.0) * (1.0 / (a * -3.0));
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.7e-247], N[(N[(b * 2.0), $MachinePrecision] * N[(1.0 / N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.70000000000000008e-247

   1. Initial program 71.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. frac-2neg 71.3%

         \[\leadsto \color{blue}{\frac{-\left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{-3 \cdot a}} \]

      2. div-inv 71.3%

         \[\leadsto \color{blue}{\left(-\left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)\right) \cdot \frac{1}{-3 \cdot a}} \]

   4. Applied egg-rr 71.3%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(c, a \cdot -3, {b}^{2}\right)}\right) \cdot \frac{1}{a \cdot -3}} \]

   5. Taylor expanded in b around -inf 61.8%

      \[\leadsto \color{blue}{\left(2 \cdot b\right)} \cdot \frac{1}{a \cdot -3} \]
   6. Step-by-step derivation

      1. *-commutative 61.8%

         \[\leadsto \color{blue}{\left(b \cdot 2\right)} \cdot \frac{1}{a \cdot -3} \]

   7. Simplified 61.8%

      \[\leadsto \color{blue}{\left(b \cdot 2\right)} \cdot \frac{1}{a \cdot -3} \]

   if 2.70000000000000008e-247 < b

   1. Initial program 21.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf 74.0%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. *-commutative 74.0%

         \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]

   5. Simplified 74.0%

      \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 67.6% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.7 \cdot 10^{-247}:\\ \;\;\;\;\frac{-0.6666666666666666}{\frac{a}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.7e-247) (/ -0.6666666666666666 (/ a b)) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.7e-247) {
		tmp = -0.6666666666666666 / (a / b);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 2.7d-247) then
        tmp = (-0.6666666666666666d0) / (a / b)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.7e-247) {
		tmp = -0.6666666666666666 / (a / b);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.7e-247:
		tmp = -0.6666666666666666 / (a / b)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.7e-247)
		tmp = Float64(-0.6666666666666666 / Float64(a / b));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.7e-247)
		tmp = -0.6666666666666666 / (a / b);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.7e-247], N[(-0.6666666666666666 / N[(a / b), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.70000000000000008e-247

   1. Initial program 71.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf 61.8%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   4. Step-by-step derivation

      1. *-commutative 61.8%

         \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]

   5. Simplified 61.8%

      \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]
   6. Step-by-step derivation

      1. *-commutative 61.8%

         \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]

      2. clear-num 61.7%

         \[\leadsto -0.6666666666666666 \cdot \color{blue}{\frac{1}{\frac{a}{b}}} \]

      3. un-div-inv 61.8%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666}{\frac{a}{b}}} \]

   7. Applied egg-rr 61.8%

      \[\leadsto \color{blue}{\frac{-0.6666666666666666}{\frac{a}{b}}} \]

   if 2.70000000000000008e-247 < b

   1. Initial program 21.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf 74.0%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. *-commutative 74.0%

         \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]

   5. Simplified 74.0%

      \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 67.6% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.7 \cdot 10^{-247}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.7e-247) (* -0.6666666666666666 (/ b a)) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.7e-247) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 2.7d-247) then
        tmp = (-0.6666666666666666d0) * (b / a)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.7e-247) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.7e-247:
		tmp = -0.6666666666666666 * (b / a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.7e-247)
		tmp = Float64(-0.6666666666666666 * Float64(b / a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.7e-247)
		tmp = -0.6666666666666666 * (b / a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.7e-247], N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.70000000000000008e-247

   1. Initial program 71.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf 61.8%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   4. Step-by-step derivation

      1. *-commutative 61.8%

         \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]

   5. Simplified 61.8%

      \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]

   if 2.70000000000000008e-247 < b

   1. Initial program 21.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf 74.0%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. *-commutative 74.0%

         \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]

   5. Simplified 74.0%

      \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.7 \cdot 10^{-247}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 42.3% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.95 \cdot 10^{+127}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 3.95e+127) (* -0.6666666666666666 (/ b a)) (* (/ c b) 0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.95e+127) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = (c / b) * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 3.95d+127) then
        tmp = (-0.6666666666666666d0) * (b / a)
    else
        tmp = (c / b) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.95e+127) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = (c / b) * 0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 3.95e+127:
		tmp = -0.6666666666666666 * (b / a)
	else:
		tmp = (c / b) * 0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3.95e+127)
		tmp = Float64(-0.6666666666666666 * Float64(b / a));
	else
		tmp = Float64(Float64(c / b) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 3.95e+127)
		tmp = -0.6666666666666666 * (b / a);
	else
		tmp = (c / b) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3.95e+127], N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.9499999999999998e127

   1. Initial program 58.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf 42.8%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   4. Step-by-step derivation

      1. *-commutative 42.8%

         \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]

   5. Simplified 42.8%

      \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]

   if 3.9499999999999998e127 < b

   1. Initial program 6.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num 6.4%

         \[\leadsto \color{blue}{\frac{1}{\frac{3 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}} \]

      2. associate-/r/ 6.4%

         \[\leadsto \color{blue}{\frac{1}{3 \cdot a} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)} \]

      3. associate-/r* 6.4%

         \[\leadsto \color{blue}{\frac{\frac{1}{3}}{a}} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \]

      4. metadata-eval 6.4%

         \[\leadsto \frac{\color{blue}{0.3333333333333333}}{a} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \]

      5. add-sqr-sqrt 0.0%

         \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(\color{blue}{\sqrt{-b} \cdot \sqrt{-b}} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \]

      6. sqrt-unprod 1.8%

         \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \]

      7. sqr-neg 1.8%

         \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(\sqrt{\color{blue}{b \cdot b}} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \]

      8. sqrt-prod 1.8%

         \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(\color{blue}{\sqrt{b} \cdot \sqrt{b}} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \]

      9. add-sqr-sqrt 1.8%

         \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(\color{blue}{b} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \]

      10. sub-neg 1.8%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\color{blue}{b \cdot b + \left(-\left(3 \cdot a\right) \cdot c\right)}}\right) \]

      11. +-commutative 1.8%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\color{blue}{\left(-\left(3 \cdot a\right) \cdot c\right) + b \cdot b}}\right) \]

      12. *-commutative 1.8%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\left(-\color{blue}{c \cdot \left(3 \cdot a\right)}\right) + b \cdot b}\right) \]

      13. distribute-rgt-neg-in 1.8%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\color{blue}{c \cdot \left(-3 \cdot a\right)} + b \cdot b}\right) \]

      14. fma-define 1.8%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\color{blue}{\mathsf{fma}\left(c, -3 \cdot a, b \cdot b\right)}}\right) \]

      15. *-commutative 1.8%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(c, -\color{blue}{a \cdot 3}, b \cdot b\right)}\right) \]

      16. distribute-rgt-neg-in 1.8%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(c, \color{blue}{a \cdot \left(-3\right)}, b \cdot b\right)}\right) \]

      17. metadata-eval 1.8%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot \color{blue}{-3}, b \cdot b\right)}\right) \]

      18. pow2 1.8%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, \color{blue}{{b}^{2}}\right)}\right) \]

   4. Applied egg-rr 1.8%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, {b}^{2}\right)}\right)} \]

   5. Taylor expanded in b around -inf 34.9%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.95 \cdot 10^{+127}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 42.2% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.95 \cdot 10^{+127}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 3.95e+127) (* b (/ -0.6666666666666666 a)) (* (/ c b) 0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.95e+127) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = (c / b) * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 3.95d+127) then
        tmp = b * ((-0.6666666666666666d0) / a)
    else
        tmp = (c / b) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.95e+127) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = (c / b) * 0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 3.95e+127:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = (c / b) * 0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3.95e+127)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	else
		tmp = Float64(Float64(c / b) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 3.95e+127)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = (c / b) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3.95e+127], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.9499999999999998e127

   1. Initial program 58.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf 42.8%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   4. Step-by-step derivation

      1. *-commutative 42.8%

         \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]

   5. Simplified 42.8%

      \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]
   6. Step-by-step derivation

      1. *-commutative 42.8%

         \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]

      2. clear-num 42.7%

         \[\leadsto -0.6666666666666666 \cdot \color{blue}{\frac{1}{\frac{a}{b}}} \]

      3. un-div-inv 42.8%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666}{\frac{a}{b}}} \]

   7. Applied egg-rr 42.8%

      \[\leadsto \color{blue}{\frac{-0.6666666666666666}{\frac{a}{b}}} \]
   8. Step-by-step derivation

      1. associate-/r/ 42.7%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666}{a} \cdot b} \]

   9. Simplified 42.7%

      \[\leadsto \color{blue}{\frac{-0.6666666666666666}{a} \cdot b} \]

   if 3.9499999999999998e127 < b

   1. Initial program 6.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num 6.4%

         \[\leadsto \color{blue}{\frac{1}{\frac{3 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}} \]

      2. associate-/r/ 6.4%

         \[\leadsto \color{blue}{\frac{1}{3 \cdot a} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)} \]

      3. associate-/r* 6.4%

         \[\leadsto \color{blue}{\frac{\frac{1}{3}}{a}} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \]

      4. metadata-eval 6.4%

         \[\leadsto \frac{\color{blue}{0.3333333333333333}}{a} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \]

      5. add-sqr-sqrt 0.0%

         \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(\color{blue}{\sqrt{-b} \cdot \sqrt{-b}} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \]

      6. sqrt-unprod 1.8%

         \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \]

      7. sqr-neg 1.8%

         \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(\sqrt{\color{blue}{b \cdot b}} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \]

      8. sqrt-prod 1.8%

         \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(\color{blue}{\sqrt{b} \cdot \sqrt{b}} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \]

      9. add-sqr-sqrt 1.8%

         \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(\color{blue}{b} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \]

      10. sub-neg 1.8%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\color{blue}{b \cdot b + \left(-\left(3 \cdot a\right) \cdot c\right)}}\right) \]

      11. +-commutative 1.8%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\color{blue}{\left(-\left(3 \cdot a\right) \cdot c\right) + b \cdot b}}\right) \]

      12. *-commutative 1.8%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\left(-\color{blue}{c \cdot \left(3 \cdot a\right)}\right) + b \cdot b}\right) \]

      13. distribute-rgt-neg-in 1.8%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\color{blue}{c \cdot \left(-3 \cdot a\right)} + b \cdot b}\right) \]

      14. fma-define 1.8%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\color{blue}{\mathsf{fma}\left(c, -3 \cdot a, b \cdot b\right)}}\right) \]

      15. *-commutative 1.8%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(c, -\color{blue}{a \cdot 3}, b \cdot b\right)}\right) \]

      16. distribute-rgt-neg-in 1.8%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(c, \color{blue}{a \cdot \left(-3\right)}, b \cdot b\right)}\right) \]

      17. metadata-eval 1.8%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot \color{blue}{-3}, b \cdot b\right)}\right) \]

      18. pow2 1.8%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, \color{blue}{{b}^{2}}\right)}\right) \]

   4. Applied egg-rr 1.8%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, {b}^{2}\right)}\right)} \]

   5. Taylor expanded in b around -inf 34.9%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.95 \cdot 10^{+127}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 10.7% accurate, 23.2× speedup

Math

\[\begin{array}{l} \\ \frac{c}{b} \cdot 0.5 \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* (/ c b) 0.5))

C

double code(double a, double b, double c) {
	return (c / b) * 0.5;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (c / b) * 0.5d0
end function

Java

public static double code(double a, double b, double c) {
	return (c / b) * 0.5;
}

Python

def code(a, b, c):
	return (c / b) * 0.5

Julia

function code(a, b, c)
	return Float64(Float64(c / b) * 0.5)
end

MATLAB

function tmp = code(a, b, c)
	tmp = (c / b) * 0.5;
end

Wolfram

code[a_, b_, c_] := N[(N[(c / b), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 49.2%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. clear-num 49.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{3 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}} \]

   2. associate-/r/ 49.1%

      \[\leadsto \color{blue}{\frac{1}{3 \cdot a} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)} \]

   3. associate-/r* 49.1%

      \[\leadsto \color{blue}{\frac{\frac{1}{3}}{a}} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \]

   4. metadata-eval 49.1%

      \[\leadsto \frac{\color{blue}{0.3333333333333333}}{a} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \]

   5. add-sqr-sqrt 36.8%

      \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(\color{blue}{\sqrt{-b} \cdot \sqrt{-b}} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \]

   6. sqrt-unprod 47.4%

      \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \]

   7. sqr-neg 47.4%

      \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(\sqrt{\color{blue}{b \cdot b}} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \]

   8. sqrt-prod 10.8%

      \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(\color{blue}{\sqrt{b} \cdot \sqrt{b}} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \]

   9. add-sqr-sqrt 30.6%

      \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(\color{blue}{b} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \]

   10. sub-neg 30.6%

       \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\color{blue}{b \cdot b + \left(-\left(3 \cdot a\right) \cdot c\right)}}\right) \]

   11. +-commutative 30.6%

       \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\color{blue}{\left(-\left(3 \cdot a\right) \cdot c\right) + b \cdot b}}\right) \]

   12. *-commutative 30.6%

       \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\left(-\color{blue}{c \cdot \left(3 \cdot a\right)}\right) + b \cdot b}\right) \]

   13. distribute-rgt-neg-in 30.6%

       \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\color{blue}{c \cdot \left(-3 \cdot a\right)} + b \cdot b}\right) \]

   14. fma-define 30.7%

       \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\color{blue}{\mathsf{fma}\left(c, -3 \cdot a, b \cdot b\right)}}\right) \]

   15. *-commutative 30.7%

       \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(c, -\color{blue}{a \cdot 3}, b \cdot b\right)}\right) \]

   16. distribute-rgt-neg-in 30.7%

       \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(c, \color{blue}{a \cdot \left(-3\right)}, b \cdot b\right)}\right) \]

   17. metadata-eval 30.7%

       \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot \color{blue}{-3}, b \cdot b\right)}\right) \]

   18. pow2 30.7%

       \[\leadsto \frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, \color{blue}{{b}^{2}}\right)}\right) \]

4. Applied egg-rr 30.7%

   \[\leadsto \color{blue}{\frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, {b}^{2}\right)}\right)} \]

5. Taylor expanded in b around -inf 8.8%

   \[\leadsto \color{blue}{0.5 \cdot \frac{c}{b}} \]

6. Final simplification 8.8%

   \[\leadsto \frac{c}{b} \cdot 0.5 \]
7. Add Preprocessing

Alternative 9: 3.6% accurate, 116.0× speedup

Math

\[\begin{array}{l} \\ -3 \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 -3.0)

C

double code(double a, double b, double c) {
	return -3.0;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = -3.0d0
end function

Java

public static double code(double a, double b, double c) {
	return -3.0;
}

Python

def code(a, b, c):
	return -3.0

Julia

function code(a, b, c)
	return -3.0
end

MATLAB

function tmp = code(a, b, c)
	tmp = -3.0;
end

Wolfram

code[a_, b_, c_] := -3.0

Derivation

1. Initial program 49.2%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. frac-2neg 49.2%

      \[\leadsto \color{blue}{\frac{-\left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{-3 \cdot a}} \]

   2. div-inv 49.1%

      \[\leadsto \color{blue}{\left(-\left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)\right) \cdot \frac{1}{-3 \cdot a}} \]

4. Applied egg-rr 49.2%

   \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(c, a \cdot -3, {b}^{2}\right)}\right) \cdot \frac{1}{a \cdot -3}} \]

5. Taylor expanded in a around 0 49.2%

   \[\leadsto \left(b - \sqrt{\mathsf{fma}\left(c, a \cdot -3, {b}^{2}\right)}\right) \cdot \color{blue}{\frac{-0.3333333333333333}{a}} \]

6. Taylor expanded in b around -inf 35.5%

   \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]

8. Simplified 3.6%

   \[\leadsto \color{blue}{-3} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024088 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))